Automated Deduction for a Multi-Modal Logic of Time and Knowledge

Introduction and logical background Multimodal logics are an appropriate framework to represent agents, actions, time, knowledge... (see [8J, [9], [14], [18], [25] for some of its uses). For example Knowledge may be represented by means of the modal operator [K] and Time by [T]; and [K]p may be read as "p is known" and [T]p as "p will be true tomorrow". A typical modal formula is [K][T]p meaning that it is known that p will be true tomorrow (which is a priori different from [T][KIp). The semantics of these logics is based in Kripke models [15], where to each modal operator is associated a binary relation over a set of so-called possible worlds. These relations have particular properties for a given modal logic system. Recently, H.J Ohlbach in [18.1, 19], A. Herzig in [10], Y. Auffray in [0], Y. Auffray and P. Enjalbert in [1] , L. Farifias del Cerro and A. Herzig in [4] have devised new proof methods for modal logics based on a translation into first order logic with specific equational theories, according to the system under concern. The way to get a proof method for a particular modal logic is, first to exhibit the associated equational theory and then to define the corresponding unification algorithm. In these papers ([4, 19, 1]) the method has been defined for monomodal logics, where only one modal operator is used. Our aim is to extend this method towards complex multimodal logics containing one or more interaction axioms, i. e. ones involving several modal operators 1. An example is the inclusion axiom [K1]p-->[K2]p, which can be read as "if agent 1 knows p then agent 2